Signal analysing and synthesizing filter bank system

ABSTRACT

In a signal analysing and synthesizing filter bank system, the analysing bank receives a signal sampled at the rate f e  and produces N contiguous subbank signals sampled at the rate f e  /N. From the subband signals the synthesizing bank must recover the incoming signal. These filter banks are formed by modulation of a prototype filter by sinusoidal signals which, for subband k (O≦k≦N=1), have a frequency (2k30 1)f e  /(4N) and respective phases +(2k+1)π/4 and -(2k+1)π/4 for the analysing and synthesizing banks. These signals are furthermore delayed by a time delay (N c  -1)/2f e ), where N c  is the number of coefficients of the prototype filter. Preferably, the analysing bank is realized by the cascade arrangement of an N-branch polyphase network (12) and a double-odd discrete cosine transform calculating arrangement (14) and the synthesizing bank is realized by the cascade arrangement of a double-odd discrete cosine transform calculating arrangement (15) and an N-branch polyphase network (17).

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a digital signal processing system comprisingan analysing filter bank for dividing an incoming signal sampled at arate f_(e) and occupying a frequency band limited to f_(e) /2, into Ncontiguous subband signals having a bandwidth of f_(e) /2N and beingsampled at the rate f_(e) /N, and a synthesizing filter bank forrecovering said incoming signal from said subband signals, said filterbanks being formed by modulating a prototype low-pass filter having afinite and symmetrical impulse response, having abeginning-of-attenuation-band frequency f_(a) less than f_(e) /2N andsatisfying the condition H² (f)+H² (f_(e) /2N-f)=1 in the band offrequencies f extending from 0 to f_(a), where H(f) is the absolutevalue of the frequency response, modulating the prototype filter beingeffected by N sinusoidal modulation signals which, for forming the k-thsubband (k extending from 0 to N-1), are characterized by the frequencyf_(k) =(2k+1)f_(e) /(4 N), by the phase α_(k) for the analysing bank andby the phase -α_(k) for the synthesizing bank, the phase α_(k) beingsuch that for the subband of the order k=0 the phase α_(o) =π/4, 3π/4,5π/4 or 7π/4 and such that for two contiguous subbands of the order kand k-1 the difference α_(k) -α_(k-1) =π/2 or -π/2.

2. Prior Art

Such systems comprising a cascade arrangement of an analysing bank and asynthesizing bank may be employed for, for example, reduced-rateencoding of a speech signal by quantizing the subband signals producedby the analysing bank with a number of variable levels depending on theenergy of each subband signal.

In order to reduce the complexity of the filters in the consideredfilter banks, the filters are given comparatively wide transition bandswhich partly overlap for contiguous filters. The result of this is thatthe recovered signal coming from the synthesizing bank can be affectedby spurious components due to the inevitable spectral folding(aliasing), which have an annoying level.

It is possible to avoid the influence of this spectral folding by anappropriate choice of the corresponding filter couples of the analysingand synthesizing banks.

This possibility has been demonstrated for filter banks having twosubbands and has been extended to filter banks having a number ofsubbands equal to a power of two, thanks to the use of special half-bandfilters: for this subject see the article by A. Croisier, D. Esteban andC. Galand, "Perfect Channel Splitting by Use ofInterpolation/Decimation, Tree Decomposition Techniques", published inthe Proceedings of the International Conference on Information Sciencesand Systems, pages 443-446, Patras, August, 1976. But, besides the factthat this procedure is limited as regards the possible number ofsubbands, it requires a comparatively large number of calculations andmemories.

For effectively realizing filter banks, there is a technique originallydeveloped in transmultiplexer systems and consisting in using for afilter bank a structure constituted by a polyphase network associatedwith a fast Fourier transform: for this subject reference is made to thearticle by M. G. Bellanger and J. L. Daguet, "TDM-FDM Transmultiplexer:Digital Polyphase and FFT", published in IEEE Trans. on Commun., Vol.COM-22, No. 9, September, 1974. It should be noted that, in thisapplication, the subbands to be realized are disjunct and the spectralfolding problem is avoided by means of filters having a narrowtransition band which is situated between the subbands.

A technique of the type used in transmultiplexer systems but suitablefor realizing the analysing and synthesizing filter banks of the systemnow under consideration, that is to say providing subbands which are notdisjunct, has been studied for certain specific cases by J. H.Rothweiler in the article "Polyphase Quadrature Filters--A New SubbandCooling Technique" published in Proc. ICASSP 83, Boston, pages1280-1283, and by H. H. Nussbaumer and M. Vetterli in the article"Computationally Efficient QMF Filter Banks" published in Proc. ICASSP84, San Diego, pages 1131-1134. For these specific cases, the number ofsubbands generally is a power of two and the number of coefficients ofeach filter of the filter bank is a multiple of the number of subbandsand, in an implementation utilizing a polyphase network and a Fouriertransform, this implies the use of a same even number of coefficients inall the branches of the polyphase network.

SUMMARY OF THE INVENTION

The present invention has for its object to avoid all these limitationsand to provide a couple of analysing and synthesizing filter banksrequiring a minimum of calculations and memories, even when the numberof subbands required is not a power of two and the number ofcoefficients required for each filter differs from a multiple of thenumber of subbands (that is to say the branches of the polyphase networkhave a variable number of coefficients).

According to the invention, the digital signal processing system, inwhich the incoming signal is sampled at the rate f_(e) and whichcomprises an analysing filter bank and a synthesizing filter bank, eachof which is formed by modulating a prototype filter by sinusoidalmodulation signals having a frequency f_(k) =(2k+1)f_(e) /(4N) and aphase +α_(k) and -α_(k) for the analysing bank and the synthesizingbank, respectively, and the phase α_(k) is such that for the subband ofthe order k=0 the phase α_(o) =π/4, 3π/4, 5π/4 or 7π/4 and such that fortwo contiguous subbands of the order k and k-1 the difference α_(k)-α_(k-1) =+π/2 or -π/2, is characterized in that said modulation signalsare furthermore delayed by a time delay τ=(N_(c) -1)/(2f_(e)), whereN_(c) is the number of coefficients required to realize thecharacteristics of the prototype filter.

In an efficient implementation of the system according to the invention,the analysing bank is formed by:

a polyphase network having N branches over which, in each period NT withT=1/f_(e), N samples of the signals entering the analysing bank aredistributed, this polyphase network being arranged for calculating, ateach period NT characterized by N and for each of its branchescharacterized by ρ (varying from 0 to N-1), a signal Pρ(m) produced fromthe coefficients of the prototype filter and from λ consecutive samplesentering the branch, where λ is the integral part of (N-N_(c) +1)/N,

a double-odd discrete cosine transform calculation arrangement receivingthe N signals Pρ(m) produced by the polyphase network and arranged forcalculating the N subband signals X_(k) (m) in accordance with theexpression: ##EQU1##

It is also advantageous to form the synthesizing filter bank in asimilar way with the aid of a double-odd discrete cosine transform,calculation arrangement, which receives the subband signals and iscoupled to a polyphase network whose output signals are time-distributedfor forming the output signal of the synthesizing bank.

BRIEF DESCRIPTION OF THE DRAWING

Features of the invention will be more fully appreciated from thefollowing description of an exemplary embodiment when considered inconjunction with the accompanying drawings, in which:

FIG. 1 shows a block diagram of the system according to the invention,in a parallel structure;

FIGS. 2a-2c show the absolute value of the frequency responses of theanalysing and synthesizing filters of the orders 0, 1, . . . , N-1;

FIG. 3 shows in FIG. 3a the absolute value of the frequency response ofthe prototype filter and in FIGS. 3b, 3c and 3d the spectra of thesignals for modulating the prototype filter, by means of which it ispossible to obtain the analysing and synthesizing filters of the orders0, 1, . . . , N-1;

FIGS. 4a-4b show a block diagram of the system according to theinvention, in a structure comprising a polyphase network and anarrangement for calculating the discrete cosine transform in order torealize each filter bank;

FIG. 5 shows an embodiment of a first portion of the analysing polyphasenetwork;

FIG. 6 shows an embodiment of the final portion of the analysingpolyphase network and the associated discrete cosine transformcalculating arrangement;

FIG. 7 shows an embodiment of the discrete cosine transform calculatingarrangement, used for realizing the synthesizing bank; and

FIG. 8 shows an embodiment of the synthesizing polyphase network.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 shows, a parallel form of the processing system according to theinvention intended for decomposing an incoming signal into contiguoussubband signals and for recovering the incoming signal from the subbandsignals, with the object, for example, of reducing the bit rate of thesignal by a suitable encoding of the subband signals.

The incoming signal x(n) sampled at the sampling rate f_(e) at instantsnT(T=1/f_(e)) and occupying a frequency band limited to the frequencyf_(e) /2, is applied to an analysing filter bank formed by a set ofparallel filters 1-0 to 1-(N-1) having impulse responses h₀ to h_(N-1).Filters 1.0 to 1-(N-1) are for dividing the frequency band (0, f_(e) /2)of the incoming signal into N contiguous subbands, each having abandwidth of f_(e) /2N. The down-sampling arrangements 2-0 to 2-(N-1)take one sample out of N samples in the signals coming from theanalysing filter bank and thus supply subband signals x_(o) to x_(N-1)sampled at the rate f_(e) /N. The down-sampling operation thus performedresults in that all the signals coming from the filters are brought downto the base-band 0-f_(e) /2N. For the sake of convenience, processingoperations effected on signals x₀ to x_(N-1), such as an adaptativequantization, are not shown. According to FIG. 1 these signals areapplied to up-sampling arrangements 3-0 to 3-(N-1) which bring thesampling rate back to the value f_(e) by adding (N-1) zeroes between twoconsecutive samples of these signals. The up-sampling thus performedresults in the spectra of the subband signals x₀ to x_(N-1) beingdistributed over the frequency axis with a repetition rate f_(e) /2. Theoutgoing signals of up-sampling arrangements 3-0 to 3-(N-1) are appliedto a synthesizing filter bank. The synthesizing filter bank is formed bya set of filters 4-0 to 4-(N-1) having respective impulse responses h'₀to h'_(N-1) so as to supply the same signals are those supplied byfilters 1-0 to 1-(N-1) of the analysing bank. Finally, the outgoingsignals of filters 4-0 to 4-(N-1) are applied to a summing arrangement 5producing an output signal x(n) which must be as accurate a replica aspossible of the incoming signal x(n).

The diagrams of FIG. 2 illustrate, by solid lines, the absolute valuesof the frequency responses of the analysing and synthesizing filterbanks for the ideal case in which these filters have a steep cut-offslope and an infinite attenuation. The diagrams 2a, 2b, 2c illustratethe respective responses H₀ (f), H₁ (f), H_(N-1) (f) of the filters 1-0,1-1, 1-(N-1) of the analysing bank and the filters 4-0, 4-1, 4-(N-1) ofthe synthesizing bank. With filters having these ideal characteristics,the outgoing signal x(n) of the signal synthesizing bank constitutes aperfect replica of the incoming signal x(n). For physically realisablefilters, having frequency responses of the type illustrated by dottedlines in the diagrams, the frequency responses of the different filtersof each bank partly overlap and the outgoing signal x(n) comprisesspurious terms due to spectral folding (aliasing).

In accordance with a prior art technique, for example the techniqueutilized in the above papers by Rothweiler and Nussbaumer, the filtersof the analysing and synthesizing filter banks are obtained bymodulating a prototype filter which, in a perfect case, has thecharacteristic of an ideal low-pass filter with a cut-off frequencyf_(e) /4, diagram 3a of FIG. 3 representing the absolute value H(f) ofthe frequency response of such a filter. The modulation signals of theprototype filter are sinusoidal and have frequencies which are oddmultiples of f_(e) /4N, that is to say (2k+1)f_(e) /(4N), where kextends from 0 to N-1. The diagrams 3b, 3c, 3d show the spectra of thesemodulation signals for the respective modulation frequencies such ask=0, k=1 and k=N-1, by means of which it is possible to obtain therespective responses H₀ (f), H₁ (f), H_(N-1) (f) shown in FIG. 2.

The prototype low-pass filter is implemented as a finite impulseresponse filter and obviously cannot have an inifite cut-off sloperepresented by the solid line response of diagram 3a. A physicallyrealizable prototype filter must satisfy several requirements, whichwill be described hereinafter, so as to ensure that the completeanalysing-synthesizing system performs its function. If f_(a) denotesthe beginning-of-attenuation-band frequency of the prototype filter, itcan be demonstrated that the spurious spectral folding terms (oraliasing terms) appearing in the outgoing signal x(n) can be eliminatedif f_(a) ≦f_(e) /2N. Diagram 3a shows by means of a dotted line theshape of the frequency response of a prototype filter satisfying thisrequirement. On the other hand, to ensure that theanalysing-synthesizing system has a unitary frequency response and alinear phase (the system being equivalent to a simple delay) theprototype filter must be chosen to be symmetrical and is to satisfy thequadrature condition:

    H.sup.2 (f)+H.sup.2 (f.sub.e /2N-f)=1                      (1)

where 0≦f≦f_(a).

On the other hand, the modulation signals of the prototype filters ofthe frequencies (2k+1)f_(e) /(4N) must satisfy the following phaserequirements, which can be derived from the above-mentioned article byRothweiler: the modulation signal producing the k-th subband must havethe phase α_(k) for the analysing bank and the phase -α_(k) for thesignal synthesizing bank, the phase α_(k) being such that for thesubband of the order k=0 the phase α_(o) =π/4, 3π/4, 5π/4 or 7π/4 andsuch that for two contiguous subbands of the orders k and k-1 thedifference α_(k) -α_(k-1) =+π/2 or -π/2. By way of example, it will beassumed hereinafter that α_(o) =π/4 and α_(k) -α_(k-1) =π/2 so thatα_(k) =(2k+1)π/4.

However, in the known systems of Nussbaumer and Rothweiler, the numberof coefficients N_(c) of the prototype filter is of necessity linkedwith the number N of the subbands by a relation of the type N_(c) =2aNor N_(C) =2aN+1, a being an integer. This constitutes a seriousrestriction of the systems, for specifically when the number N ofsubbands is high, as, in order to satisfy one of these relations, one isgenerally forced to use for the prototype filter a number ofcoefficients N_(c) which is higher than strictly necessary to obtain therequired filtering characteristitics. On the other hand, these knownsystems are only described practically for the case in which the numberN of subbands is a power of 2, whereas for example a number of subbandssuch as 12 seems to be well-suited for subdividing the 4 kHz band of atelephone signal. All things considered, the prior art signal analysingand synthesizing systems are not flexible and very often do not allow anoptimum reduction of the number of calculations and memories.

The present invention permits of obviating these disadvantages by anoptimum handling of the case of a number of subbands differing from apower of two with a number of coefficients of the prototype filterindependent of the number of subbands.

According to the invention, the modulation signals are furthermoredelayed by a time τ=(N_(c) -1)/(2f_(e)).

Thus, taking account of all the requirements necessary for themodulation signals (frequency, phase and delay) and taking account ofthe fact that these signals are sampled at the rate f_(e) (samplinginstants n/f_(e)), one can easily see that the n-th coefficient h_(k)(n) of the k-th filter of the analysing bank and the n-th coefficienth'_(k) (n) of the k-th filter of the synthesizing bank can be derivedfrom the n-th coefficient h(n) of the prototype filter, in accordancewith the expressions: ##EQU2##

In these expressions (2) and (3), the above-mentioned case where α_(k)=(2k+1)π/4 has been taken by way of example. On the other hand, theseexpressions are valid for the N values of k extending from 0 to N-1 andfor the N_(c) values of n extending from 0 to N_(c) -1.

These formulae show more specifically that the coefficients h_(k) (n)and h'_(k) (n) of the analysing and synthesizing filters depend on thenumber N_(c) of coefficients of the prototype filter, this number Nchaving any possible value and being determined more specifically so asto satisfy the requirements as regards the steepness of the cut-offslope. In known systems, this number Nc is linked to the number N by theabove-defined relations and does not appear in the known correspondingformulae (see for example said article by Nussbaumer, formula (1)). Putdifferently, it can be said that, in the known systems, the filtersdetermined by the coefficients h_(k) (n) and h'_(k) (n) must each besymmetrical (have a linear phase), whereas there is no such requirementin the system according to the invention, provided that each couple ofcascade-arranged filters h_(k) (n) and h'_(k) (n) behaves as alinear-phase filter.

Now it will be demonstrated that the system according to the inventionshown in FIG. 1, provided with analysing and synthesizing filters haverespective coefficients h_(k) (n) and h'_(k) (n) defined by the formulae(2) and (3), performs its function by substantially eliminating thespectral folding terms (aliasing terms) and by substantially behaving asa pure delay.

To that end, first the equations linking the input and output signals ofthe system x(n) and x(n), as a function of the filters having thecoefficients h_(k) (n) and h'_(k) (n), will be described.

The calculations are effected in the Z-domain where:

    Z=e.sup.-2πjfT with j.sup.2 =-1, T=1/f.sub.e.

It should be noted that: W=e⁻²πj/N.

In accordance with common practice, the transformed variables in theZ-domain are written with capital letters.

The signal x_(k) of the k-th subband produced by the analysing bank isobtained by filtering the input signal with the aid of the filtershaving the coefficients h_(k) (n), followed by a down-sampling operationby a factor 1/N with the aid of an arrangement 2-k. This down-samplingoperation conveys the useful signal of the k-th subband to the baseband(-f_(e) /2N, +f_(e) /2N), but also all the imperfectly filteredcomponents of the signal which represent the spectral folding terms(aliasing term). In the Z-domain, this signal x_(k) can be written as:##EQU3##

The value u=0 of the variable u provides the contribution of the usefulsignal in the signal of the subband k as conveyed to the baseband. Thevalues u=1 to N-1 provide the contributions of the spurious componentsaliased to the baseband and coming from other subbands.

After an up-sampling operation by a factor of N by adding zeroes in anarrangement, 3-k, the transformed signal of the subband becomes:##EQU4##

The signal of the subband k, filtered by the synthesizing filter havingcoefficients h'_(k) (n), is then written in the transform domain as:##EQU5##

The Z-transform of the signal x(n) coming from the synthesizing bank canthen be written: ##EQU6##

To ensure that the overall analysing-synthesizing system is equivalentto a pure delay DT, the condition must be satisfied that X(Z)=Z⁻³.X(Z),that is say in accordance with (5): ##EQU7##

From the relations (2) and (3), which give the expressions of thecoefficients h_(k) (n) and h'_(k) (n), it is easy to derive that theirrespective Z-transforms can be written as: ##EQU8##

These expressions for H_(k) (Z) and H'_(k) (Z) can be transferred to theexpression (4) for G_(u) (Z).

Now first the case in which u=0 will be described. By replacing W and Zin the expression (4) by their respective values, it can be demonstratedthat ##EQU9## where H(f) is the absolute value of the frequency responseof the prototype filter. It is easy to check that the quadraturecondition of the prototype filter defined in formula (1), renders itpossible to satisfy the relation: ##EQU10## whatever the value of k maybe.

From this it follows that:

    G.sub.0 (f)=e.sup.-2πjf(N.sbsp.c.sup.-1)T

which implies that the useful value of the output signal X(Z) is theincoming signal X(Z), delayed by a time DT=(N_(c) -1)T, which is equalto double the delay produced by the prototype filter having N_(c)coefficients.

Now the case will be examined in which u=1 to N-1, for which it will bedemonstrated that G_(u) (Z)=0, which means that in the output signalX(Z) the spurious components due to spectral folding are eliminated.

Taking account of the relations (7) and (8), the expression (4) forG_(u) (Z) can be written as: ##EQU11##

Now there appear terms of the type H(W^(p/4) Z).H(W^(q/4) Z) whichcorrespond to the products of the two filters derived from the prototypefilter, one by a translation over pf_(e) /4N and the other by atranslation over qf_(e) /4N. Assuming the attenuation of the prototypefilter to be sufficiently high (at least 40 dB), the terms may beneglected for which p and q are such that there are no overlapping zonesbetween H(W^(p/4) Z) and H(W^(q/4) Z), inclusive of the passbands andtransition bands. Only those terms of the type H(W^(p/4) Z).H(W^(q/4) Z)subsist for which p and q are such that overlap occurs, that is to saysuch that k=u and k=u-1 (which corresponds to two contiguous subbands).It is then possible to demonstrate that the last expression for G_(u)(Z) can be simplified as follows: ##EQU12##

By replacing the terms φ_(u), ψ_(u), φ_(N-u) ψ_(N-u), etc. by theirvalues given in (9) for u=k, it is found that G_(u) (Z)=0 for 1≦u≦N-1,which is the relation to be verified so as to ensure that the spectralfolding terms are cancelled in the outgoing signal of the synthesizingbank.

In practice, it has been possible to realize an analysing-synthesizingsystem which divides the 4 kHz band of a speech signal, which is sampledat 8 kHz, into 12 subbands having widths f_(e) /2N=333 Hz with aprototype filter having a number of coefficients N_(c) =66 for obtainingan attenuation of 3 dB at the frequency f_(e) /4N=166 Hz and anattenuation increasing to over 40 dB above the frequency f_(e) /2N=333Hz, the quadrature condition (1) being satisfied in the band from 0 to333 Hz.

Now it will be shown how the signal analysing and synthesizing filtersof the system according to the invention can be implemented in aparticularly effective manner, in accordance with the block diagramsshown in FIG. 4. In accordance with block diagram 4a representing theanalysing bank, the incoming signal x(n) sampled at the rate f_(e) isapplied to a commutator circuit 10 which cyclically distributes thesamples of x(n) over N branches 11-ρ (ρ extending from 0 to N-1), sothat the sampling rate in each branch is reduced to f_(e) /N). The Nsignals in the branches 11-ρ are applied to a polyphase analysingnetwork 12, at whose input the respective signal samples are assumed tobe adjusted to the correct phase and in which they are subjected to aprocessing operation which will be explained hereinafter. The N outgoingsignals of the branches 13-ρ of the polyphase network 12 are applied toa processing arrangement 14, which calculates a double-odd discretecosine transform as will be described in greater detail hereinafter. TheN outgoing signals of the processing arrangement 14 constitute thedown-sampled outgoing signals (sample rate f_(e) /N) of the analysingbank which are denoted by x₀ to x_(N-1) in FIG. 1. In diagram 4a thesesignals are designated for convenience by X_(k) (m), k extending from 0to N-1 characterizing the subband of a signal and m characterizing thesampling instant.

As shown in diagram 4b representing the synthesizing bank, the Noutgoing signals X_(k) (m) of the signal analysing bank are applied to aprocessing arrangement 15, which calculates a double-odd discrete cosinetransform which will be described in greater detail hereinafter. The Noutput going signals of the processing arrangement 15 are applied toinput branches 16-ρ of a synthesizing polyphase network 17 in which theyare subjected to a processing operation which will be described ingreater detail hereinafter. The N signals appearing at the outputbranches 18-ρ of the polyphase network 17 are sampled at the reducedrate f_(e) /N and have samples which may be assumed to be in-phase inthe N branches. These N signals are applied to the respective N inputsof a commutator circuit 19, which distributes the respective signalsamples in the time so as to form at its output a signal sampled at therate f_(e) which constitutes the outgoing signal x(n) of the signalsynthesizing bank.

The following description explains how the structures 4a and 4b renderit possible to realize the respective operations to be effected in thesignal analysing and synthesizing banks.

This description will first be given for the analysing bank. At eachinstant mNT the signal X_(k) (m) of the subband k formed by the signalanalysing bank results from a convolution between the N_(c) coefficientsh_(k) (n) of the filter determining this subband and N_(c) consecutivesamples of the signal entered in the analysing bank up to the instantmNT. Consequently, this signal X_(k) (m) can be written as: ##EQU13##where E(x) represents the integral part of x.

In the expression (10) it is possible to effect a change in thevariables permitting to determine the variable n by two other variablesr and ρ, such that:

    n=rN+ρ

where ρ varies from 0 to N-1 and r varies from 0 to λ-1.

It should be noted that with the complete excursion of the variables rand ρ, the variable n varies from 0 to λN-1. If, as the presentinvention permits, the number N_(c) of the coefficients h_(k) (n) isless than λN, the values of h_(k) (n) for n going from N_(c) to λN-1must be chosen equal to zero, these zero values being symmetricallydistributed at the two ends of the impulse response h_(k) (n) implyingthat λN-N_(c) is even.

With this change in the variables and by using for the coefficientsh_(k) (n) the values given by the expression (2), expression (10) forX_(k) (m) becomes: ##EQU14##

This formula (11) can be written differently by using the followingnotations: ##EQU15##

With these notations it is possible to demonstrate that the expression(11) can be written in the following form, where λ and N are even:##EQU16##

By using the following properties of the function C_(N) (k,ρ): ##EQU17##it is possible to demonstrate that the expression (15) can be written:##EQU18##

Now the results will be given which are obtained in the cases ofdifferent artities for λ and N.

•λ even and N odd: ##EQU19## and where P.sub.ρ.sup.(1) andP.sub.ρ.sup.(2) are given by the formulae (14).

•λ odd, any parity for N: ##EQU20##

The remaining part of the description of the signal analysing bank willbe given by way of example for the case where λ and N are even.

Thus, the calculations to be effected in the analysing bank in order toobtain the sub-band signals X_(f) (m) can be divided in two distinctportions:

In a first filtering portion, the formulae (17) completed with theformulae (14) are used for calculating the signals P.sub.ρ (m) from thefiltering coefficients h.sub.ρ (r) and samples x.sub.ρ (m-r) of theincoming signal of the analysing bank, h.sub.ρ (r) and x.sub.ρ (m-r)being defined in the formula (12). This first portion of thecalculations is realized in the analysing polyphase network 12 shown inFIG. 4a;

In a second modulation portion which puts the formula (16) into effect,the signals P.sub.ρ (m) are modulated with the aid of a double-odddiscrete cosine transform having the coefficients defined in the thirdformula (12). This second stage is realized in the arrangement 14.

Now a block diagram for carrying out the calculations effected in thepolyphase network 12 for forming the signals P.sub.ρ.sup.(1) (m) andP.sub.ρ.sup.(2) (m) will be described with reference to FIG. 5. Thisblock diagram is given for the above-mentioned case where the number Nof subbands to be realized is N=12 and the number N_(c) of coefficientsof the prototype filter is N_(c) =66. For these values of N and N_(c),the parameter λ defined in formula (10a) has the value 6. Thus, thevariables ρ and r defined in the foregoing vary

    for ρ: from 0 to 11

    for r: from 0 to 5.

The block diagram of FIG. 5 comprises the commutator circuit 10 alreadyshown in FIG. 4 which cyclically distributes the samples of the inputsignal x(n) over 12 branches 11-ρ(ρ varying from 0 to 11) constitutingthe inputs of the polyphase network 12. A shift register 20 is used in awell-known manner for re-adjusting the samples distributed over the 12input branches to the proper phase and thus supplies in the branches21-ρ the signals x.sub.ρ (m) whose samples are in-phase and are producedat the instants mNT. The block diagram show only the arrangementconnected to the branch 21-ρ for processing the signal x.sub.ρ (m) andfor forming the signals P.sub.ρ.sup.(1) (m) and P.sub.ρ.sup.(2) (m) inaccordance with the calculations shown in formula (14), it beingunderstood that similar arrangements must be connected to the 12branches 21-0 to 21-11 for forming these signals for all values of ρ.

The branch 21-ρ is connected to a delay line constituted by circuits22-1 to 22-5, each of which produces a delay NT (represented by thefunction Z^(-N)). The samples x.sub.ρ (m) to x.sub.ρ (m-5) are availableat the input of the delay line and at the output of the circuits 22-1 to22-5, respectively, that is to say the samples x.sub.ρ (m-r) (r varyingfrom 0 to 5) required for applying the formulae (14). These samplesx.sub.ρ (m) to x.sub.ρ (m-5) are applied to respective inputs ofmultipliers 23-0 to 23-5 to be multiplied by the respective coefficients-h.sub.ρ (0), -h.sub.ρ (1), h.sub.ρ (2), h.sub.ρ (3), -h.sub.ρ (4),-h.sub.ρ (5). It can be verified that -h.sub.ρ (1), h.sub.ρ (3),-h.sub.ρ (5) are the only non-zero terms obtained by having r vary from0 to 5 in the factor h.sub.ρ (r)• cos (2r-λ)π/4 of the first formula(14). It can also be verified that -h.sub.ρ (0), h.sub.ρ (2), -h.sub.ρ(4) are the only non-zero terms obtained by having r vary from 0 to 5 inthe factor -h.sub.ρ (r)• sin [(2r-λ)π/4] of the second formula (14). Allthese non-zero terms are derived from the coefficients of the prototypefilter h.sub.ρ (r) defined in the foregoing.

Applying the first formula (14), the products formed by the multipliers23-1, 23-3 and 23-5 are added together with the aid of adders 24-3 and24-5 and the sum thus obtained constitutes the desired value of thesignal P.sub.ρ.sup.(1) (m). By applying the second formula (14), theproducts formed by the multipliers 23-0, 23-2 and 23-4 are addedtogether with the aid of adders 24-2 and 24-4 and the sum thus obtainedconstitutes the desired value of the signal P.sub.ρ.sup.(2) (m).

It should be noted that the structure of the polyphase network shown inFIG. 5 having a total of 12 branches which are similar to thoseconnected to the branch 21-ρ, renders it possible to use a prototypefilter having up to 72 coefficients (λ•N=72). In the present case, inwhich this number of coefficients of the prototype filter is 66, thereare 6 coefficients h.sub.ρ (r) used in the polyphase network which willhave zero values, these zero-value coefficients which correspond to thetwo ends of the impulse response of the prototype filter being: h₀(0)=h₁ (0)=h₂ (0)=h₉ (5)=h₁₀ (5)=h₁₁ (5)=0.

The block diagram shown in FIG. 6 is given for the same case as theblock diagram of FIG. 5: N=12, N_(c) =66. It comprises the final portionof the polyphase network 12 provided with 12 output branches 13-ρ andused to form in accordance with the formula (17) the 12 signals P.sub.ρ(m) from the signals P.sub.ρ.sup.(1) (m) and P.sub.ρ.sup.(2) (m), whichare calculated in a manner as described with reference to FIG. 5. Theblock diagram of FIG. 6 includes thereafter the double-odd discretecosine transform calculation arrangement 14 to which the 12 signalsP.sub.ρ (m) are applied and which has for its object to form the 12desired sub-band signals X_(k) (m).

The block diagram of FIG. 6 shows on the left 24 branches to which the12 pairs of signals P.sub.ρ.sup.(1) (m) and P.sub.ρ.sup.(2) (m) areapplied. The adders 30-0 to 30-5 receive the signals indicated in FIG. 6for forming, in accordance with the first formula (17), the 6 signalsP_(o) (m) to P₅ (m) based on the 12 signals P₀.sup.(2) (m) toP₁₁.sup.(2) (m). The subtractors 30-6 to 30-11 receive the signalsindicated in FIG. 6 for forming, in accordance with the second formula(17), the 6 signals P₆ (m) to P₁₁ (m) on the basis of the 12 signalsP₀.sup.(1) (m) to P₁₁.sup.(1) (m).

In the arrangement (14) for calculating the discrete cosine transformutilizing formula (16), the 12 signals P₀ (m) to P₁₁ (m) available atthe output branches 13-ρ of the polyphase network 12 are applied torespective multipliers 31-0 to 31-11 in order to be multiplied by therespective coefficients C_(N) (k, 0) to C_(N) (k, 11). Thesecoefficients are determined with the aid of the third formula of the setof formulae (12) and depends on the number k of the subband. By the oddfactors (2k+1) and (2ρ+1) figuring in this formula, the coefficientsC_(N) (k, ρ) are characteristic of a double-odd discrete cosinetransform. The adders 32-1 to 32-11 are connected in such a way as toform the sum of the products produced by the multipliers 31-0 to 31-11,this sum constituting the signal X_(k) (m) of the subband k, inaccordance with formula (16). All the subbands signals X₀ (m) to X₁₁ (m)to be produced by the analysing bank are formed in the same manner, byemploying the coefficients C_(N) (k, 0) to C_(N) (k, 11) where k variesfrom 0 to 11.

Now it will be described how the synthesizing bank can be realised inaccordance with the structure of FIG. 4b.

The output signal x(n) of the synthesizing bank may be considered asbeing the result of the interleaving of N down-sampled signals x(mN+ρ),where m characterizes the sampling instants produced at the rate f_(e)/N and ρ, which varies from 0 to N-1, characterizes each down-sampledsignal. The following description will basically determine theoperations to be effected for obtaining each down-sampled signalx(mN+ρ), denoted as x.sub.ρ (m).

To determine these operations, it is assumed, just as in the analysingbank that n=rN+ρ, where r varies from 0 to λ-1, λ being defined asindicated in formula (10a).

At each instant mNT, a signal x.sub.ρ (m) results from the sum of the Nsignals which are each formed by filtering a subband signal X_(k) (m)with the aid of the λ coefficients of a synthesizing filter defined byits coefficients h_(k) (n). To put it more accurately, a signal x.sub.ρ(m) can be calculated in accordance with the expression: ##EQU21## whereX_(k) (m-r) represents λ samples produced in the subband k up to theinstant mNT and h'_(k) (rN+ρ) represents an assembly characterized by anumber ρ of the λ coefficients of the synthesizing filter of the subbandk.

By utilizing for the coefficients h'_(k) (rN+ρ) the values given by theexpression (3) and by utilizing the notations in formulae (12), it canbe demonstrated, using calculations similar to those effected for theanalysing bank that the expression (18) can be written in the followingform for the case in which λ and N are even: ##EQU22##

By making use of the symmetry of the prototype filter h(n) expressed byh(r)=h_(N-1-)ρ (λ-r-1), the expression (20) can be written: ##EQU23##

Now the results will be given which are obtained in the case, in whichthe parities of λ and N are different.

•λ even and N odd: ##EQU24## •λ odd, any parity of N: ##EQU25##

As was also the case for the analysing bank, the following part of thedescription of the synthesizing bank will be given for the case in whichλ and N are even.

Thus, the calculations to be effected in the synthesizing bank in orderto obtain the output signal x(n) can be divided in two distinctportions:

In a first modulation portion utilizing formula (21), the subbandsignals X_(k) (m) are modulated with the aid of a double-odd discretecosine transform for forming the signals Y.sub.ρ (m). This first portionis realized in the calculation arrangement 15 of FIG. 4b.

In a second filtering portion, the formulae (19) and (20) or (22) areutilized for calculating the signals x.sub.ρ (m) from the filteringcoefficients h.sub.ρ (r) and signals y.sub.ρ (m) produced by the firstportion. The second portion is realized in the polyphase synthesizingnetwork 17 of FIG. 4b. The output signal x(n) is obtained by thetime-distribution of the signals x.sub.ρ (m).

Now a block diagram of an embodiment of the arrangement 15 forcalculating the double-odd cosine transform will be described withreference to FIG. 7 for the said case where N=12 and λ=5, so that thevariable ρ varies from 0 to 6 and the variable r varies from 0 to 11.

In accordance with FIG. 7, the arrangement 15 has 12 input branchesreceiving the respective subband signals X₀ (m) to X₁₁ (m). These 12subband signals are applied to respective multipliers 40-0 to 40-11 inorder to be multiplied there by the coefficients C_(N) (0, ρ) to C_(N)(11, ρ) determined by the third of said formulae (12). Adders 41-1 to41-11 are connected in such a way as to form the sum of the signalsproduced by the multipliers 40-0 to 40-11, this sum constituting thesignal y.sub.ρ (m) determined in accordance with formula (21). By usingthe coefficients C_(N) (0, ρ) to C_(N) (11, ρ) for ρ varying from 0 to11, 12 signals y_(o) (m) to y₁₁ (m) are obtained. It should be notedthat the arrangements 14 and 15 of the analysing and synthesizing filterbanks have the same structures and employ the same set of coefficientsC_(N) (k, ρ) which are characteristic of a double-odd discrete cosinetransform.

The block diagram of FIG. 8 effects in the polyphase network 17 thecalculations defined by the expressions (19) and (22) for the case inwhich N=12 and λ=6. The use of the formulae (19) and (22) renders itpossible to obtain for the same value of ρ going from 0 to N/2-1 (0 to5), the two signals x.sub.ρ (m) and x_(N-1-)ρ (m).

The block diagram of the polyphase network 17 given for the currentvalue of ρ (varying between 0 and 5) has two inputs 50 and 51 receivingfrom the arrangement 15 the respective signals ##EQU26## A delay lineformed from 5 elements each producing a time delay NT is connected tothe input 50 and a delay line formed by 4 elements each producing a timedelay NT is connected to the input 51. At the points b₁, b₃, b₅ of thedelay line connected to the input 50 there are the respective samples##EQU27## At the points b₀, b₂, b₄ of the delay line connected to input51 there are the respective samples ##EQU28##

The points b₁, b₃, b₅ are connected to respective inputs of multipliers52-1, 52-3, 52-5. The other input of these multipliers receives thecoefficients -h.sub.ρ (1), h.sub.ρ (3), -h.sub.ρ (5) which represent thenon-zero terms h.sub.ρ (r)• cos [(2r-λ)π/4] appearing in formula (19),for r varying from 0 to λ-1, i.e. 0 to 5. The multipliers 52-1, 52-3,52-5 produce products which are added together with the aid of adders53-3, 53-5 for producing the first sum S₁ which renders it possible tocalculate x.sub.ρ (m) in accordance with the formula (19), i.e.:##EQU29## Moreover, the points b₀, b₂, b₄ are connected to therespective inputs of multipliers 52-0, 52-2, 52-4. The other input ofthese multipliers receives the coefficients h.sub.ρ (0), -h.sub.ρ (2),h.sub.ρ (4), which represent the non-zero terms h.sub.ρ (r)• sin[(2r-λ)π/4], appearing in formula (19), for varying from 0 to λ-1, i.e.from 0 to 5. The multipliers 52-0, 52-2, 52-4 produce products which areadded together with the aid of adders 53-2, 53-4 for producing thesecond sum S₂ which renders it possible to calculate x.sub.ρ (m) inaccordance with formula (19) i.e.: ##EQU30##

An adder 54 forms the sum S₁ +S₂ which constitutes the signal x.sub.ρ(m) appearing at the output branch 18-ρ of the synthesizing polyphasenetwork.

On the other hand the samples ##EQU31## appearing at the points b₅, b₃,b₁ of the delay line connected to input 50 may also be represented by##EQU32## Similarly the samples ##EQU33## appearing at the points b₄,b₂, b₀ of the delay line connected to input 51 may also be representedby ##EQU34## The points b₅, b₃, b₁ are connected to an input ofmultipliers 55-5, 55-3, 55-1 whose other input receives the coefficients-h.sub.ρ (0), h.sub.ρ (2), -h.sub.ρ (4). The products produced by themultipliers are added together with the aid of adders 56-3, 56-5 forproducing the signal S'1. It can be easily verified that S'1 is thefirst sum rendering it possible to calculate x_(N-1-)ρ (m) in accordancewith formula (22), i.e.: ##EQU35## On the other hand the points b₄, b₂,b₀ are connected to an input of multipliers 55-4, 55-2, 55-0, whoseother input receives the coefficients -h.sub.ρ (1), h.sub.ρ (3),-h.sub.ρ (5). The product produced by these multipliers are addedtogether with the aid of adders 56-2, 56-4 for producing the signal S'2.It can be easily verified that S'2 is the second sum rendering itpossible to calculate x_(N-1-)ρ (m) in accordance with formula (22),i.e.: ##EQU36##

A subtractor 57 produces the difference S'1-S'2 which constitutes thesignal x_(N-1-)ρ (m) appearing at the output branch 18_(N-1-)ρ of thesynthesizing polyphase network.

The calculations of x.sub.ρ (m) and x_(N-1-)ρ (m) are realized in thesynthesizing polyphase network for N/2 values of ρ going from 0 to N/2-1(thus 6 values of ρ going from 0 to 5), which renders it possible toobtain at the N=12 output branches of the polyphase network 18-0 to18-11 the down-sampled signals x₀ (m) to x₁₁ (m) which aretime-distributed with the aid of the commutator circuit 19 for producingthe output signal x(n) of the synthesizing bank.

With the overall signal analysing and synthesizing bank system, eachprovided with a polyphase network and an arrangement for calculating thediscrete cosine transform, the number of multiplications per seconddecreases to ##EQU37## that is to say to 140,000 for N=12 and N_(c) =66and the number of additions per second decreases to ##EQU38## that is tosay to 124,000.

What is claimed is:
 1. A digital signal processing system comprising:a.an analysing filter bank for dividing an incoming signal, sampled at arate f₃ and occupying a frequency band limited to f_(e) /2, into Ncontiguous subband signals having respective bandwidths of f_(e) /2N andwhich are sampled at a rate f_(e) /N, said analyzing filter bankcomprising N prototype low-pass filters, one for each subband andarranged in parallel along N respective branches, each prototypefilter:i. having a same finite and symmetrical impulse response; ii.having a beginning-of-attenuation-band frequency f_(a) less than f_(e)/2N; and iii. satisfying the condition H² (f)+H² (f_(e) /2N-f)=1 in aband of frequencies f extending from 0 to f_(a), where H(f) is theabsolute value of the frequency response to each prototype filter; andiv. comprising a modulation input for receiving a modulation signal,each modulation signal having:A. a frequency f_(k) =(2k+1)f_(e) /(4N),where k is an integer, running from 0 N-1, representing a respective oneof the N subbands; B. a phase α_(k), such that α₀ is equal to one ofπ/4, 3π/4, 5π/4, and 7π/4, and such that for two contiguous subbandsrepresented by k and k-1, the difference α_(k) -α_(k-1) is equal to oneof π/2 and -π/2; and C. a time delay τ=(N_(c) -1)/(2f_(e)), where N_(c)is a number of coefficients required for realizing the prototype filter;and b. a synthesizing filter bank for recoverying said incoming signalfrom said subband signals, said synthesizing filter bank comprising Nprototype low-pass filters which are the same as the prototype filtersused in the analyzing filter bank, except that the phase of themodulation signals is -α_(k), whereby the number N is any integergreater than or equal to 2 and whereby N_(c) is not constrained to be amultiple of N.
 2. A digital signal processing system comprising ananalysing filter bank for dividing an incoming signal, sampled at a ratef_(e) and occupying a frequency band limmited to f_(e) /2, into Ncontiguous subband signals X_(k) (m) having respective bandwidths off_(e) /2N and each being sampled at a rate of f_(e) /N, the analyzingfilter bank comprising a polyphase network which comprises:a. means fordistributing N samples of the incoming signal over N branches in eachperiod NT, with T=1/f_(e) ; b. first means for calculating N signalsP.sub.ρ (m) from λ consecutive signals entering the ρth branch and froma set of coefficients, said first calculating means having N inputs forreceiving the samples from the distributing means, where ρ is an integervarying from 0 to N-1, λ is the integer part of (N-N_(c) +1)/N, andN_(c) is the number of coefficients in the set; c. N second means forcalculating a double-odd discrete cosine transform, each secondcalculating means having N inputs for receiving the signals P.sub.ρ (m)and having an output at which a subband signal X_(k) (m) is supplied,said subband signal being formed according to the following formulae:i.for λ odd and for λ even and N even: ##EQU39## ii. for λ even and N odd:##EQU40## whereby the analyzing filter bank filters an incoming signalaccording to a frequency response which is given by a succession of Nelementary frequency responses, each of the N frequency responses havinga shape which corresponds to a frequency response to a same prototypefilter, which prototype filter is defined by the N_(c) coefficients andsupplied with a modulation input for receiving a plurality of signalshaving respective frequencies and whereby N is any integer greater thanor equal to two and whereby N_(c) is not constrained to be a multiple ofN.
 3. A system as claimed in claim 2, characterized in that firstcalculating means calculates each signal P.sub.ρ (m) in accordance withthe following formulae:(a) for λ even and N even: ##EQU41## h.sub.ρ (r)representing λ coefficients h(n) of the prototype filter for n=rN+ρ,x.sub.ρ (m-r) representing λ samples entered into a branch ρ of thepolyphase network up to said period NT characterized by m.
 4. The systemof claim 2 or 3 comprising a synthesizing filter bank for recoveringsaid incoming signal from said subband signals, the synthesizing bankcomprising:(a) N third means for calculating a double-odd discretecosine transform, having N inputs for receiving the N subband signalsX_(k) (m) and having N outputs at which N signals Y.sub.ρ (m) areprovided according to the following formulae: for λ odd and for λ evenand N even: ##EQU42## (b) a polyphase network having N inputs forreceiving the signals Y.sub.ρ (m) and comprising fourth means forcalculating, at each period NT, indicated by m, and for each branch,indicated by ρ, a signal x.sub.ρ (m), from the coefficients of theprototype filter and from λ consecutive samples of Y.sub.ρ (m); (c)means for time distributing the N samples of x.sub.ρ (m) to produce arecovered input signal.
 5. A system as claimed in claim 4, wherein thefourth calculating means calculates the signal x.sub.ρ (m), inaccordance with the following formulae:(a) for λ even and N even:##EQU43## (b) for λ even and N odd: ##EQU44## (c) for λ odd: ##EQU45##